Free carrier absorption in III nitride semiconductors

Transcription

1 Chapter 5 Free carrier absorption in III nitride semiconductors 5.1 Introduction The absorption of electromagnetic radiation, due to its interaction with electrons in semiconductors, is essentially determined by two distinct processes: interband and intraband transitions. The electronic transitions may be direct or indirect according as the band extrema occur at the same k or different k points in the Brillouin zone. The minimum photon energy for interband absorption defines the fundamental absorption edge. In the second process, namely, free-carrier absorption (FCA), electrons in the conduction band and holes in the valence band are excited to higher energy states in the same band. FCA, is essentially an intraband indirect transition phenomenon. It accounts for the absorption of electromagnetic radiation of frequencies lower than those which give rise to interband transitions and necessitates the mediation of phonons or imperfections to conserve Part of the work presented in this chapter has appeared in Journal of the Physical Society of Japan, 82, (2013) AIP Conf. Proc., 1391, 72 (2011) Proc. of 15 th, International Workshop on Physics of Semiconductor Devices, XV, 579 (2009) 99

2 overall momentum. In the presence of phonons or imperfections a free carrier may absorb a photon via a second order process in which the carrier changes its momentum by scattering. FCA, intrinsically connected to scattering, is therefore a powerful means of determining the possible scattering mechanisms operative and understanding transport properties in a system [5.1]. In the last few years III nitrides and their alloys have attracted attention as potential material systems for use in high speed opto electronic devices such as blue light-emitting diodes (LEDs), blue lasers and solar blind ultraviolet photodetectors [5.2, 5.3]. The commonly studied nitride based compound semiconductors are GaN, InN and AlN whose room temperature minimum band gaps range from 0.7 ev for InN through 3.4 ev for GaN to 6.2 ev for AlN [5.3]. Among these, GaN, because of the large direct band gap and its chemical and physical stabilities, has emerged as an attractive candidate for devices operating at high temperatures, high voltages and high power at microwave frequencies. However, nitride semiconductor material systems, including GaN, InN and AlN are known to be characterized by large built-in piezoelectric fields as well as the presence of unintentional imperfections such as threading dislocations introduced during growth process, due to lattice mismatch between substrate and active material. Threading dislocations can be of the screw, edge and mixed types. Edge dislocations formed at the substrate-buffer interface with mainly vertical orientation thread to the epilayer surface whereas the number of screw and mixed dislocations are known to decrease with distance from the interface. The order of edge dislocations is found to range from 10 7 cm -2 to cm -2 [5.2, 5.3]. However, in nitrides, the dislocations parallel to the c axis do not couple any piezoelectric potential [5.4]. These edge dislocations are found to limit carrier transport properties considerably in nitride 100

3 quantum well systems [5.4] and are expected to influence performance of optical devices as well. The influence of dislocations on the optical absorption spectra of nitrides is found to manifest in significant loss coefficients, and in enhancement of optical absorption near fundamental absorption edge and a red shift of the absorption edge [ ]. In GaN and InGaN systems, Kioupakis and coworkers [5.11] employing the first principles approach, and considering scattering by phonons, charged-defect and alloy scattering, have found FCA an important loss mechanism. The peculiarities of FCA in InN epitaxial layers with wide range of electron concentrations have been investigated by Nargelas et al [5.12] and Wu et al [5.13]. With a view to estimate the influence of dislocations on the FCA in nitride systems, we have, in this chapter, made a study of FCA first in 2D QW systems and then in bulk nitride systems. In section 5.3, we present our theory, developed for the first time, for FCA assisted by dislocation scattering via Coulomb and strain fields in quantum wells. The absorption of radiation by a free carrier is treated by a second-order perturbation in which the interaction of the 2DEG with dislocations and with the radiation field is considered simultaneously. Investigations of FCA in bulk nitrides are lacking. In section 5.4, we present a systematic analysis of FCA coefficient in bulk GaN data considering electrons to be scattered by acoustic phonons via deformation and piezoelectric interactions, impurities, optical phonons and dislocations. 5.2 Theory The FCA coefficient, K, is given by [5.1], (5.1) 101

4 where κ, is the dielectric constant of the medium, n o the number of photons in the radiation field and f i the carrier distribution function. The sum is over all initial states i of the system. W + i and W - i represent the transition probabilities for the absorption and emission of photons, respectively and can be calculated using the standard second order Born golden rule approximation. These are defined by. (5.2) In (5.2) the transition matrix elements are given by, (5.3) where H rad is the electron-photon interaction Hamiltonian and V D the electron-dislocation scattering potential. ħω is photon energy and E i, E j and E f denote the initial, intermediate and final state energies of electrons, respectively. The sum is over all the intermediate states j of the system. The expression for K can be evaluated using the wavefunction, eigenvalues and the appropriate distribution function. The total FCA coefficient, K, due to various scattering mechanism (s) is given by [5.1]:. 5.3 Free carrier absorption in nitride quantum wells The edge dislocations are known to grow normal to the plane of the quantum well. The electrons in quantum well are scattered by the dislocations via strain and coulomb interactions. The scattering from the long range strain field surrounding the dislocations may be via deformation and piezoelectric potentials. In the case of 2D systems, FCA has been theoretically studied extensively in GaAs/AlGaAs structures [ ]. There are no systematic studies of FCA due to 102

5 dislocation scattering even in nitride quantum structures except for the investigation of Wu et al in InN layers [5.13]. We consider a 2DEG in a square QW system with the electrons assumed to be confined to move in the x-y plane. For simplicity, we consider the QW to be of infinite depth. The electron wave functions and energy eigen values are given by [5.14], n = 1, 2 (5.4) and, (5.5) where, is position vector, is electron wave vector, d is width of the QW, n is the subband index and. For a non-degenerate quasi 2D electron gas, the distribution function can be expressed as [5.14], (5.6) with,, and n s is the sheet carrier concentration. Assuming electromagnetic radiation to be polarized along the plane of the QW the matrix elements of the electron photon interaction Hamiltonian can be expressed as [5.14], (5.7) where, e is unit vector in the direction of polarization of the radiation field Expression for dislocation-assisted FCA coefficient FCA, which is intrinsically associated with carrier scattering through second order process, requires a quantitative description of the dislocation scattering processes in a 2D 103

6 system. The scattering of the carriers by the edge dislocations in these systems is due to the coulomb potential produced by the charges on the dislocation lines and the long-range strain field surrounding the dislocation lines [5.4]. Dislocation scattering of 2DEG in QW systems has been studied by many workers, considering the interaction of electrons with the strain fields surrounding edge dislocations via deformation potential coupling [5.17, 5.18] and with charged dislocations via coulomb potential [5.18]. Since the electric fields generated by dislocations do not extend over large distances and are rather localized around the core of dislocations, the screening of the interaction is weak. In the present work we consider the interactions to be unscreened Dislocation scattering via strain field The effect of strain field around dislocations is to shift the conduction and valence band edges. The perturbing potential for electron scattering can be expressed as [5.17], (5.8) where θ is the polar angle with respect to Burgers vector, b along QW plane, q is in-plane wave vector, a c, the conduction band offset and γ the Poisson ratio. Using eqns. (5.4) and (5.8) the expression for the matrix elements of electrondislocation scattering via strain interaction is expressed as, (5.9) where, and is the angle between b and q. The screening function with. In the absence of screening. 104

7 Using (5.1), (5.4), (5.6) and (5.9), we obtain the following expression for FCA coefficient due to scattering by dislocations (of density N d ), via unscreened strain field where, (5.10), dislocation density., and is exponential integral function and N d is Dislocation scattering via Coulomb interaction Modelling a threading dislocation, growing perpendicular to the QW plane, as a line of charge with charge density ρ L, we obtain the expression for interaction potential of electrons with charged line dislocations as, (5.11) where κ w and κ b are the dielectric constants of the material in well and barrier, respectively. Assuming. κ w = κ b and using (5.11), we obtain the expression for the matrix element of electron-dislocation scattering via coulomb field as (5.12) Here,,, and. 105

8 Using (5.1), (5.4), (5.6) and (5.12), we obtain expression for FCA coefficient due to scattering by dislocations (of density N d ) via unscreened coulomb interaction as. (5.13) Here and Results and Discussion We have performed numerical calculations of FCA coefficient, K using equations (5.10) and (5.13) considering scattering of electrons by edge dislocations. Here, we present the results for three nitride quantum well systems of GaN/AlGaN, InN/AlN and AlN/AlGaN. The material parameters, characteristic of GaN, AlN and InN, used in our calculations are given in table 5.1. Table 5.1:Material parameters of GaN, InN and AlN [5.3] Parameter GaN InN AlN Effective mass, m * (m o ) Conduction band offset, a c (ev) Burgers vector, b (Ǻ) Poisson ratio, γ Dielectric constant, ε Lattice constant, c (Ǻ)

9 We choose to illustrate the behavior of FCA for quantum wells of width d = 100 Ǻ and carrier concentration n s = 1 x cm -2. The dislocation line charge density,, is taken as where f is the fraction of filled acceptor states and c o, the lattice spacing in the (0001) direction. We take N d = 10 8 cm -2, and assume f =1. Figures 5.1, 5.2 and 5.3 depict respectively, the frequency dependence of the dislocation mediated FCA coefficient in GaN/AlGaN, InN/AlN and AlN/AlGaN quantum well systems at T = 300K. In each of the figures 5.1, 5.2 and 5.3, curves a and b represent the contributions to FCA from the strain field and the Coulomb field, respectively. Curves 1 denote the overall contributions. The following common features are noticed. For the parameters and range of frequencies considered, the FCA coefficient decreases with increase in photon frequency,. A kink is observed whenever the photon frequency equals corresponding to transition to the second subband. It may be noted that, this kink occurring along with the FCA process is peculiar to QW THz systems. This is incontrast to the bulk case where, FCA and interband transitions occur as separate processes. The dominant contribution to overall dislocation-mediated FCA is due to scattering via the strain field of the dislocations. In the case of GaN QW, the large contribution to the total FCA (curve 1) from the strain field (curve a) may be due to large conduction band deformation potential as compared to that of InN and AlN. It is also noticed that, the position of the kink shifts towards higher (lower) energy region for InN (AlN), as compared to that in GaN. This shift may be due to the varying values of the effective masses of electron in the three systems (see table I). The value of the FCA coefficient is found to be enhanced at the subband transition energy with the increase in 107

10 K ( 10 3 cm -1 ) K (10 3 cm -1 ) 100 GaN 10 1 a b (10 13 s -1 ) Figure 5.1: Variation of dislocation mediated FCA coefficient, K, as function of photon frequency,, for the GaN quantum well of width d = 100 Ǻ, carrier concentration n s = 1 x cm -2 and dislocation density N d = 1 x 10 8 cm -2. Curves a and b depict K due to dislocation scattering via strain and coulomb interactions, respectively. Curve 1 represents the total K. 12 InN a b ( x s -1 ) Figure 5.2: Variation of dislocation mediated FCA coefficient, K, as function of photon frequency,, for the InN quantum well of width d = 100 Ǻ, n s = 1 x cm -2 and N d = 1 x 10 8 cm -2. Curves a and b represent K due to dislocation scattering via strain and coulomb interactions, respectively. Curve 1 depicts the total K. 108

11 K ( 10 3 cm -1 ) AlN a b ( x s -1 ) Figure 5. 3: Variation of dislocation mediated FCA coefficient, K, as function of photon frequency,, for the AlN quantum well of width d = 100 Ǻ, n s = 1 x cm -2 and N d = 1 x 10 8 cm -2. Curves a and b depict K due to dislocation scattering via strain and coulomb interactions, respectively. Curve 1 represents the total K. absorption coefficient near the kink found to be larger for InN compared to that of AlN and GaN. A comparison of the behavior of FCA in QWs due to scattering of 2DEG by dislocation strain with that in bulk systems [5.13] shows the magnitudes of K to be larger in QW system. This may be because the increased localization of charge within the 2DEG, could enhance scattering of the 2DEG [5.17]. We have also investigated the influence of QW width and dislocation density on FCA. In figure 5.4, curves 1, 2 and 3 show variation of FCA coefficient in GaN QWs with N d =10 8 cm -2 for well widths, d = 125 Ǻ, d = 100 Ǻ and d = 75 Ǻ, respectively. It may be noted that, the position of kink at redshifts with increase in well width. This is because the 109

12 Figure 5.4: Variation of dislocation mediated FCA coefficient, K as function of photon frequency,, for the GaN quantum well of n s = 1 x cm -2. Curves 1, 2 and 3 depict total K for N d = 1 x 10 8 cm -2 well widths, d = 125 Ǻ, d=100 Ǻ and d = 125 Ǻ, respectively. Curves 2, 4 and 5 depict the variation of total K with dislocation density, N d = 1 x 10 8 cm -2, N d = 1 x 10 9 cm -2 and N d = 1 x cm -2, respectively for QW of width d = 100 Ǻ. confined states of the well are closer in energy for larger well widths. Our results are consistent with experimental studies of Bayram in AlGaN/GaN superlattices [5.19]. Curves 2, 4 and 5 depict the variation of FCA coefficient for the GaN QW of width d = 100 Ǻ for dislocation densities, N d = 1 x 10 8 cm -2, N d = 1 x 10 9 cm -2 and N d = 1 x cm -2, respectively. An increase in number of dislocations results in an increase in FCA. This follows from eqns. (5.10) and (5.13), indicating that larger loss coefficient can result from higher dislocation densities [5.6]. It may be noted that a similar dependence is exhibited by 2DEG scattering rate [5.4, 5.18], a first order scattering process. In conclusion, we have developed a theory of edge dislocation assisted FCA in QWs assuming the scattering via coulomb and strain fields. Calculations of frequency dependence 110

13 of FCA coefficients in QWs of three nitride systems GaN,InN and AlN are presented. The dominant contribution to the FCA is found to be due to the strain field of the dislocations. The frequency dependence of FCA coefficients exhibits kinks whenever the frequency corresponds to transition to the second subband. The position of kinks shows redshift with increase in quantum well width. The theory shows larger loss (absorption) coefficients due to increase in dislocation density [5.7]. It may be mentioned here that, the inclusion of the screening of the interactions is expected to reduce the FCA coefficients. An estimate of the effect of screening for large angle scattering (q = 2k F ;, being Fermi wavevector) in the QWs of GaN, InN and AlN indicates a reduction in the FCA coefficients by approximately 20%, 13% and 42% respectively. It may be mentioned that, with proper control of the parameters characterizing dislocations, one may obtain information about the band structure of the QW system especially in those regions of the subbands might be explored which cannot be reached by the electrons in an experiment on transport phenomena. 5.4 Free carrier absorption in bulk nitrides Light absorption in nitrides arises because of scattering of free carriers from inevitable acoustic phonons, via deformation potential and piezoelectric interaction, optical phonons and unintentional impurities, and dislocations introduced during growth process. In particular, FCA depends on the individual contributions due to various scattering mechanisms operative in the system. Here, we are interested in investigating the influence of dislocation scattering on FCA in bulk nitrides. In literature, there exist, investigations of FCA in bulk semiconductors with regard to dislocation scattering [5.6, 5.7]. However, the role and importance of the contribution from 111

14 dislocation scattering to FCA in GaN seems to be unclear. Cunninghaum et al [5.20] have measured room temperature IR spectra for 3 GaN samples in the region of 1 < λ < 3.5μm. Their measurements showed a characteristic FCA described by a wavelength dependence of s absorption coefficient, with s ranging from 2.2 to 3.9, characteristic of optical mode(s ~ 2.5) and impurity (s ~ 3.5) scatterings. Vignaud and Farvacque [5.6] proposed that the lowenergy component of optical absorption in GaAs induced by strong electric fields resulting from charged dislocations, may be observed only at low temperature. Ambacher et al [5.21] used photothermal deflection spectroscopy to study the sub-band gap absorption of GaN thin films in the range eV. They correlate the FCA below 1.5eV with electron concentration and find s ~ 2. Hasegawa et al [5.22], who have recently measured optical absorption spectra for plastically deformed n-gan, have observed, in the long wavelength region (λ > 1μm), a decrease in FCA by deformation. The electron wavefunction and eigenvalues for a bulk semiconductor are (see (1.2) and (1.3)) (5.14) and. (5.15) The distribution function for a non-degenerate electron gas is (5.16) The FCA coefficient K, for bulk semiconductor can be obtained using expressions (5.1), (5.2), (5.3) and (5.16) as [5.23] 112

15 (5.17) where,,, and being photon and phonon energies, respectively, and. In (5.17) the sum is over all scattering mechanisms, s, and, represents the matrix elements for electron phonon, electron impurity, or electron dislocation scattering interaction Hamiltonian. The expressions for the electron-radiation and electron-imperfection scattering interaction Hamiltonians are documented in literature [5.23, 5.24] Analysis of FCA data in bulk GaN We have performed numerical calculations of FCA coefficient, K using (5.17) for parameters characteristic of bulk GaN (see table 5.1) [5.1] at T=300K in range of wavelength, λ, 1 50μm. The other parameters used are typical of the GaN sample of Hasegawa [5.22]: impurity concentration N i = 5 x m -3, dislocation density N d = m -2, and optical phonon energy, ħω q = 91 mev. 113

16 K (m -1 ) 10 6 d e c b 10 3 a ( m) Figure 5.5: Variation of FCA coefficient, K as a function of wavelength of incident radiation for bulk GaN. Curves a, b, c, d and e respectively show individual contribution to FCA due to polar optical phonons, dislocations, acoustic phonons via deformation, piezoelectric couplings, and impurities. Curve 1 represents total contribution. Figure 5.5 depicts variation of FCA coefficient as a function of wavelength, λ of the incident radiation. Curves a, b, c, d and e represent the variation of FCA coefficient due to scattering electrons by polar optical phonons, dislocations, acoustic phonons via deformation and piezoelectric couplings, and impurities, respectively. Curve 1 represents the total contribution to FCA. We find FCA increases with increase in the wavelength of incident radiation. Dependence of absorption coefficient K, on λ is found to be s ~ 2.3. For the parameters chosen, absorption due to acoustic phonon scattering via both deformation and piezoelectric couplings is dominant in the range of wavelength, 1 5μm. Absorption due to impurity scattering becomes important at wavelengths, λ > 5μm. Effect of dislocations on FCA is minimal. For the wavelength, λ > 30μm, absorption due to polar optical phonons becomes important. 114

17 K (m -1 ) c b a ( m) Figure 5.6: Variation of FCA coefficient in bulk GaN at 300K for three impurity concentration: m -3 (curve a), m -3 (curve b) and m -3 (curve c), m -3. In figure 5.6, curves a, b, and c show variation of FCA coefficient for impurity concentration 10 19, and 10 25, m -3 at T = 300K respectively. For the parameters considered, we find that, an increase of impurity concentration from to m -3 increases, the absorption by one order of magnitude at λ = 1μm. The effect of impurity concentration is large for higher wavelengths. Figure 5.7 shows a comparison of our numerical results of FCA coefficient with the measured data of Hasegawa [5.22] in GaN in the range of wavelengths 1 2 μm. For the parameters considered a good fit is obtained. For the GaN sample considered we take impurity concentration, N i = 5 x m -3 and dislocation density N dis = m -2 [5.2]. Curves a e denote individual contributions to FCA coefficient due to scattering of electrons from acoustic 115

18 K (mm -1 ) a b 1 0 c d e ( m) Figure 5.7: Variation of FCA coefficient in bulk GaN. Curves a, b, c, d and e represent the individual contribution due to acoustic deformation, piezoelectric, impurity, dislocation and polar optical phonons scatterings, respectively. Curve 1 denotes total contribution. Circles denote experimental data of [5.22]. phonons via deformation, and piezoelectric, impurities, dislocations and polar optical phonons, respectively. Curve 1 represents total contribution. Circles denote measured data. K increases with increase in wavelength of incident radiation. Dependence of K on wavelength is 2.3 and it agrees with experiment. K, is found to be of the order of few mm -1. We find that, for the range of wavelengths the considered, contribution from acoustic phonon scattering via deformation potential coupling is large compared to that from piezoelectric phonon and impurity scatterings. The wavelength dependence for acoustic phonons via deformation potential and piezoelectric couplings is same. FCA due to dislocation and polar optical phonon scattering is small. It is also found that, the effect of impurity scattering is more at higher wavelengths. 116

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